Forgot to write, if parts are left on the PCB, then a full block can be reused, either by cutting the traces and rewiring it, or simply by cutting out the PCB around the block to be reused. For example power regulator ICs with nearby resistors and filtering capacitors, audio (pre)amplifiers IC with nice ground plane PCB and proper layout, and so on.
Sometimes it works cutting simply with a pair of snipers, or with metal sheet cutting scissors.
When cutting the PCB with power tools, a respiratory mask or other method for avoiding inhaling fiberglass dust should be used.